Interest rate modelling II term structure models (1)

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Part III

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10

One-Factor Short Rate Models I

So far, our focus has been on vanilla models suitable for simple securities for which a change of measure allows the price to be expressed as an expectation of (a function of) a single random variable, typically a forward swap or Libor rate However, many practically important securities, such as those that are callable or path-dependent, depend on interest rates in a substantially more complex manner, necessitating the construction of models for the dynamics of the entire discount curve — and not just a select few points on it We have already, in Chapter 4, outlined the HJM theory that governs all dynainic cliscount curve models driven by vector-valued Brownian motions The general HJM class with its infinite-dimensional Markovian dynamics is, however, too unwieldy to work with in practice, so it is of considerable interest to identify HJM model sub-classes that involve a finite number of Markov state variables only We shall devote several chapters to this task, covering first the “classical” approach of writing down an explicit SDE for

the short rate r(t)

In our treatment of short rate models, we start out in this chapter with an in-depth analysis of the one-factor mean-reverting Gaussian model, providing a classical perspective on a model that we encountered in a modern HJM setting in Chapter 4 The chapter also covers the affine one-factor model, of which the Gaussian model is a special case In Chapter 11, we generalize our cliscussion to arbitrary one-factor SDEs for the short rate, and finally, in Chapter 12, we introduce the class of multi-factor short rate models

For derivatives pricing purposes, the short rate modeling approach has largely been superseded by newer approaches Still, short rate models remain quite popular in empirical work, and a good understanding of these models provides a strong foundation for work with more sophisticated models

10.1 The One-Factor Gaussian Short Rate Model

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408 10 One-Factor Short Rate Models I

P(t,T) = E? (= fyi re) #) | (10.1)

so knowledge of the risk-neutral dynamics for r(t) is in principle sufficient to compute time ¢t discount bond prices to all maturities J’ > t In practice, the expectation in (10.1) may, of course, not be computable in closed form, so to make short rate models operational in practice we must look for the sub-class of models where (10.1) is either analytically tractable or, at the very least, amenable to fast numerical methods

One approach for which (10.1) becomes particularly tractable is to model the short rate as a Gaussian random variable The resulting Gausszan short rate (GSR) model has a long and distinguished history in the financial literature While our applications focus leaves us little room for historical ruminations, we shall make a slight concession here, by developing the GSR model progressively from the historically important — yet ultimately impractical — special case in Ho and Lee [1986] Our development of the model will also initially progress by classical “bottoms-up” means, developing the dynamics of the forward curve from an SDE for the short rate, rather than the other way around Besides providing some historical perspective, our style of presentation involves several generally applicable techniques and should give the reader additional intuition about the mechanics of the models involved

10.1.1 The Ho-Lee Model

10.1.1.1 Notations and First Steps

Starting from the fundamental assumption that the short rate r(t) is adapted to a single Brownian motion W(t), the simplest possible dynamics we can

imagine is the martingale process r(t) = r(0) + 0,W(t), or

dr(t) = 0, dW(t), (10.2)

where a, > 0 is a constant and W(t) is a Brownian motion in the risk-neutral measure Q From the basic risk-neutral pricing relationship (10.1), the time

¢ discount bond maturing at time T then must have the price

P(t,T) = E, G F7 r(w) *) = E, (e-Œ=9~zr LEW) du) (10.3)

where ky = EP is the time ¢ risk-neutral expectation operator

Lemma 10.1.1 If r(t) follows (10.2) in the risk-neutral measure, then

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10.1 The One-Factor Gaussian Short Rate Model 409

Proof We notice that + r(u) = r(t) ff a,dW(s), tu >t, t so that _ / * rude = —r(t)(T —t) — op [ / ˆ đW(3) dụ The order of integration can be changed by Fubini’s theorem (see Duffie [2001]), such that (Ƒ dW(s) umf ƒ eatts = (ứ-s dW (3) By the Ito isometry, it then follows that — fr r(u) du is Gaussian with mean —r(t)(T —t) and variance Var, [re = H2 [ (T —s)? ds = 202/7 — £)°

The result of the lemma then follows from basic moment properties of

log-normal variables, see e.g (1.22) O

Let us define a yield y(t, 7) = —In P(t,T)/(T —t), such that

y(t,T) =r(t) — <02(T — 42

The yield curve shapes that can be produced by the simple model in (10.2) are rather primitive, as is evident from this expression In particular, the yield curve is always downward-sloping in T —t and yoo = limp_5 y(t, 7) = —G© 10.1.1.2 Fitting the Term Structure of Discount Bonds

The model presented above effectively has only two parameters — r(0) and go, ~~ with which one can attempt to fit the initial yield curve It should be clear that this is insufficient to properly match observable discount bond prices, which effectively disqualifies the model from practical pricing applications Fortunately, as realized in the paper Ho and Lee [1986], a remedy is quite straightforward!: simply introduce a deterministic function

a(£) and alter the model to be

r(t)=r(0)+ a(t) +o,W(t), a(0) =0, (10.4)

‘The original paper by Ho and Lee was set exclusively in discrete time ‘Lhe continuous-time version of the model developed here is, we feel, significantly more

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such that

dr(t) = a(t) dt + o, dW(t), (10.5) where a’(t) is the first-order derivative of a(t) To match the discount bond curve at time 0, a(t) cannot be freely stipulated, but must be set as specified in Lemma 10.1.2 below

Lemma 10.1.2 Let r(t) be given as in (10.4), and assume that discount

bond prices at time 0, P(0,T), are known for all T > 0 Set

a(t) = f(0,t)—r(0) +2022, f(0,t) 3 = 2B POD ôi

Then, for any T > 0, E G foo ru) m) = P(0,T) (10.6) Proof Applying Lemma 10.1.1, we get t E G Jo re) *) = exp (—r(oye + sort’ | x exp (~ / a(u) au 0 from which it follows that (10.6) is satisfied if : i -| a(u) du = In P(0,t) + r(0)t — ort 0 Taking derivatives with respect to t yields In P a(t) = ae) ~ (0) + £04? = f(0,t) — r(0) + sont? L]

The model (10.4) with a(t) set as in Lemma 10.1.2 is known as the Ho-Lee

model We characterize the model further in the following proposition Proposition 10.1.3 In the Ho-Lee model, the risk-neutral process for r(t) is dr(t) = (a + ot) dt + 0,dW(t), (10.7) and bond prices at time t can be reconstituted from r(t) through the expression P(0,T) P(0,t) 1

P(t,T) = exp (~(r()~ f10,8))(T-#) - Zo2e er?)

Proof Equation (10.7) follows directly from (10.5) when a(t) satisfies Lemma

10.1.2 ‘To show the second part of the proposition, applying Lemma 10.1.1

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10.1 The One-Factor Gaussian Short Rate Model All P(t,T) = exp Gay —a(t))\(T —t)+ sơ — 0°) x exp § [ atu) 2 = exp Gao —a(t))(T —t) + 272(T ~ 99) 1 x exp (1 P(0,7) + r(0)T — G01” — In P(0,) — r(0)£ + sort’) P(0,T) Ì_2/m,2 _ m2

¬ P0, exp ( —Œ() (r(t) — a(t) — r(0))(T —t) + 5 or (Tt — a(t) ~r(0))(T —t) + = _ T*t) |

In this expression —a(t) — r(0) = —f(0,t) — 02t?/2 from the definition of a(t) The result follows O

10.1.1.3 Analysis and Comparison with HJM Approach

Lo gain a better understanding of the Ho-Lee model, let us examine the dynamics for bonds and forward rates implied by the model From Proposition 10.1.3, we get f(t,T) = aT) = f(0,T) +r(t)- f(0,t)+o7t(T—t) (10.8) and af (t,T) = dr(t) + Go — 23) — Ot | dt = 02(T —t)dt +0,dW(t) (10.9) In similar fashion, we get adP(t,T)/P(t,T) = r(t) dt — o,(T — t) dW(t)

In the notations of Section 4.4, we have thus established that forward

rate volatilities in the Ho-Lee model are ø/(f,T) = ơ„ and discount bond volatilitles are ơ p(f, 1) = øz(T'—t) Due to the constancy of ø;(£, 7), random

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zero in linear fashion as t > T, reflecting the pull to par phenomenon discussed earlier in Chapter 4

Setting aside for a moment the question about whether the Ho-Lee model is a reasonable representation of the real world, let us make a brief interlude to point out that we could, in fact, have specified the model directly as an HJM model with of(t,T) =o, and a single Brownian motion The HJM result, Lemma 4.4.1, then immediately establishes the drift in the SDE for

f(t, T) to be

+

urít,T) = a | o, du = 07 (T —t), t

consistent with (10.9) above Integrating this equation establishes (10.8), from which the discount bond reconstitution formula in Proposition (10.1.3) follows To establish (10.7), we simply write r(t) = f(t,t) and differentiate:

9| ay oars (2029 + ot) ay

dr(t) = df(t,T)|rat + Ot

T=t

where the second equality uses (10.8)—(10.9) Notice that arriving at Propo- sition 10.1.3 in this manner did not involve evaluation of any expectations The Ho-Lee model] has several drawbacks that disqualifies it for most, if not all, pricing applications We list some of them below

e The constancy of forward rate volatilities as a function of forward rate maturity (7 — t) is unrealistic: long-dated forward rates are less volatile than short-dated ones

e The constancy of forward rate volatilities as a function of calendar time t gives the model time-stationary dynamics, but also results in the model having far too few degrees of freedom to allow for calibration to quoted option prices

e Spot and forward interest rates are Gaussian and can therefore become negative, which is unrealistic

e The model has only one driving Brownian motion and instantaneous moves of all forward rates are therefore perfectly correlated, contrary to empirical evidence

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10.1 Vhe One-Factor Gaussian Short Rate Model 413

10.1.2 The Mean-Reverting GSR Model 10.1.2.1 The Vasicek Model

Many empirical studies find that interest rates exhibit mean reversion, in the sense that if an interest rate is high by historical standards, it will most likely fall in the future (and vice versa if the interest rate is low) To

model this phenomenon, Vasicek [1977] assumed that the short rate follows

a one-factor Ornstein- Uhlenbeck process in the risk-neutral measure:

dr(t) = (0 — r(f)) dt + 0, dW (t), (10.10)

where x,U0,o, are positive constants From results for the linear SDE in Section 1.6, it follows that the short rate can be written t r(t) = 0+ (r(0) — Be" + vãi e~Z~%) 3W(a) (10.11) 0 It follows that r(t) is a Gaussian random variable with moments E(r(t)) = 0+ (r(0) — Ve" *, (10.12) 2 o

Var ar (r(é)) = SE (1 e2) (r(t)) = == (1-e° 7") (10.13) 10.13

As t — oo, the mean of the short rate approaches Ở and the variance goes to

o?/(22) Accordingly, 0 is often known as the long-term level (or sometimes

the mean reversion level) of the short rate ‘The speed at which the short rate can be expected to revert to its long-term level is determined by »*, known as the mean reversion speed

To establish a discount bond pricing formula in the Vasicek model, we use (10.11) to write 7 I r(u)du = —0t — (r(0) — 9) (1— e””?) /z t u — Ơy / / es) JW(s) du 0 20 Clearly — Jo r( u) du is Gaussian, with mean —%t£ — (r{0) — 9) (1 — e 7") |x

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414 10 One-Factor Short Rate Models [

From the usual result for log-normal variables, it follows that discount bond prices in the Vasicek model can be computed as

P(0,1) =exp ự (- | *(u) au + 2 Var (- / 'fa) iu)

1—e 0

=exp(T=——Ê—r(0)— exp ( r(0) + “(L—e~* + = e )

Ơ 2

X exp lấn (-er?: + 4eT? + 2‡zz— 3)

More generally, we have the following proposition, the proof of which is straightforward Proposition 10.1.4 Define _ —x{(T-t) B(t,T) =——— ơ2 2B t,T)? A(t,T) = (0 _ Z| (B(t, T) ~ (T —t)) - —- Then, in the Vasicek model (10.10), P(t, T) = exp (A(t, 7) — Bt, T)r(t))

As we did for the model in Section 10.1.1.1, define y(#,7) = —In P(t, T)/(P —t), and notice that now a finite limit exists,

Yo = lim y(t,T) = 9 —o2/ (2x7)

T- 0°

In the Vasicek model, three different yield curve shapes are possible

Lemma 10.1.5 Let y(t,T) = —InP(t,T)/(f —t) Then If r(t) > 0, then y(t,T) decreases in T —t

If r(t) < Yoo — 02/(427), then y(t,T) increases in T — t

Otherwise, y(t, T') first increases in T —t and then decreases (i.e y(t, T) is humped)

Proof By straightforward, but tedious, calculus O

While this is certainly an improvement over the martingale model we encountered in Section 10.1.1.1, the Vasicek model is still not capable of fitting the observable yield curve accurately enough for pricing applications It should be obvious that the way to solve this problem is to mimic the step that lead to the Ho-Lee model in Section 10.1.1.2: introduce a deterministic function of time into the definition (10.11) That is, we write

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10.1 Vhe One-Factor Gaussian Short Rate Model 415

where a(t) is a deterministic function and zog(£) is the short rate in the

Vasicek model The function a(t) is determined from the condition that

B (er non6)48) 2= 4094 — 0,2),

where the right-hand side is assumed given Further development of this model proceeds as in Section 10.1.1.2, and results are easily imagined; we skip the analysis as the resulting model is a special case of the more general setup in Section 10.1.2.2 below We do note, however, that the Vasicek model — both with and without adjustment to fit the initial yield curve — is easily

shown to have forward rate and discount bond volatilities of

1— e— #(P-t) )

+

2;(1,T) =ø,e"~Đ,— øp(t,T) = ø (

Introduction of mean reversion into the model will thus introduce expo- nential decay in the term structure of forward rate volatilities From an empirical standpoint this is considerably more appealing than the maturity- independent forward rate volatilities in the Ho-Lee model, and also in qualitative agreement with the fact that short- and medium-maturity interest rate options trade at higher implied volatilities? than do long-dated options While this is a step up from the Ho-Lee model, the model still has too few degrees of freedom for many derivatives pricing applications, as the model will rarely calibrate well to observed prices of vanilla options (e.g European swaptions and caps) We improve on this in the next section

10.1.2.2 The General One-Factor GSR Model

The most general form of the one-factor GSR model is given by the SDE

dr(t) = x(t) (8(t) — r(t)) dt +0,(t) dW(t), (10.14)

i.e we have now allowed all parameters in the Vasicek moclel to depend on time While this model can be developed by classical means (see e.g Hull and White [1994a] for, often laborious, details), it is significantly easier to work within an HJM setting In fact, we already showed in Section 4.5.2 that

short rate dynamics of the form in (10.14) must originate from a “separable”

HJM model of the form

df (t,T) = o;(t,T) (/" a p(t, w) a dt + ơr(t,T) ÄdW), (10.15)

T

o f(t, T) = o,(t) exp (- | s(t) ts)

Chapter 4 also proved the following result for the function v(t)

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Proposition 10.1.6 For the general one-factor GSR model (10.14) to match the initial yield curve, we must have 1 df(0,t) x(t) Ot + f(0,¢) + t v(t) = 1 [ en 2 Sus) sq (ay)? du, x(t) Jo

Proof Follows from Proposition 4.5.4, whend=1 O

We notice the presence of Of(0,t)/Ot in the expression for Ở(£) (a similar

term was, of course, present in the Ho-Lee model) which can be a nuisance in applications where the initial forward curve is not smooth, as when we have used simple bootstrapping to construct the curve To get rid of the term,

we now switch variables, from r(t) itself to z(t) = r(t) — f(0,t) Dynamics for z(t), as well as the bond reconstitution formula for (10.14) in terms of x(t) are listed next Proposition 10.1.7 Define x(t) = r(t) — f(0,t) Then, for the model (10.14)-(10.15), dz(t) = (y(t) — x(t)x(t)) dt + 0,(t)dW(t), z(0) =0, (10.16) where , — en? fi xe(s)ds u)2 du y(t) [ (u)* d (10.17)

The bond reconstitution formula is

P(t,T) = Fon exp (-z0e0, 7)— sy(t)đ(t,7)7) (10.18) T

G(Œ, 1`) = / ew Je" 248) 48 doy, t

Proof To simplify notation, define K(t) = f x(u)du, and set g(t) = o,(t)e®™, A(t) =e"*K™., Then oy (t,T) = g(t)h(T) and, by integration of

(10.15),

f(t,T) = f(0,T) + A(T) | g(u)? / h(s) ds du + h(T) | g(u) dW(w)

" ° (10.19)

Set

x(t) =n | a(u? | h(s) isdu+n(e) | g(u) dW (wu),

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=J'{£) ([ se” [ h(s) ds du) dt + h(t)? ( fot)? du) dt + h(t)g(t)dW(t )+'@) | g(u) dW (u) dt = tớ +z(9) dt + h(t)g(t) dW(t) = (y(t) — x(t)x(t)) dt + o-(t) dW(t), where , = ft 2 +( 2 +4 y(t) = h(t) Lai 2d was defined in (10.17) From (10.19) we have f(t, T) = f(0,T) + ty t)+ h' ) [a6 rf h(s) ds du hữ) [ su [ hö) s) ds du = (0,7) + man! vt mà ng (91 [ t@ [mova , f(t, t

such that in particular r(t) = f(t,t) = f(0,t) + x(t), as claimed earlier Inserting the expression for f(t, T) an the basic relation T P(t, T) = exp (-ƒ f(t, u) produces (10.18) after a few rearrangements O = f(0 Remark 10.1.8 The discount bond dynamics for P(t,T) are dP(t,T)/P(t,T) =r(t)dt —op(t,T)dW(t), op(t,T) =0,(t)G,T) Remark 10.1.9 In the reconstitution formula (10.18), notice that G(t,T) = (G(0,T) — G(0, t)) eo #48,

a result that is often useful in grid-based numerical work (see Section 10.1.5)

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418 10 One-Factor Short Rate Models |

10.1.2.3 Time-Stationarity and Caplet Hump

A Gaussian HJM model is said to be time-stationary if the instantaneous volatility a(t, 7) is only a function of 7'— ứ, i.e the time to maturity rather than the tine of maturity 7 Time stationarity is an appealing feature, as it implies that the volatility term structure of forward rates will look the sane in the future as it does today; in the absence of other information, this prediction is often very reasonable and in good agreement with empirical observation In the setting of the one-factor GSR model, imposing time- stationarity will require us to set both a,(t) and x(t) to constants, such that

ơr(t,T) =ơ„e Z—9, (10.20)

In other words, the only thae-stationary forward rate volatility term structure that can be constructed in the GSR model is an exponentially decaying one In practice, however, it is quite common to observe (from the caplet market, say) forward rate volatility structures that have a marked “hump”, with short-dated options trading at very low volatilities This effect can largely be attributed to central bank activity, as the extreme short end of the forward curve tends to move primarily in response to central bank changes to funding rates As such changes are relatively infrequent and normally quite predictable*, short-dated forward rates are typically associated with relatively little uncertainty and, consequently, have low volatilities

If we attempt to match a GSR model to a humped forward volatility structure, it follows from (10.20) that this cannot be done in a stationary manner and we are forced to let % become a function of time ‘To see this,

suppose that we at time 0 observe forward volatilities a/(0,T) = 6(T), where b(T’) is a humped function of 7, i.e b(7) initially increases in T but

ultimately decreases in T Ideally, we would like to set of (t, 7) = b(T' — ‡), but this is not possible in the GSR setting, as explained above To inake

the GSR model match 6(7) at time 0, we instead are forced to make x a

function of time, determined from the relation

ơz(0,1') = o,e7 lộ z()đ — b(T), oa, = b(0)

Taking logarithins and differentiating gives

_ a(n b(é)) _ ote)

z4) = dt b(t)”

If t, is the time t at which b(t) reaches its peak (i.e b’(t,) = 0), it follows

that x(t) will be negative for all t < t, and positive for all t > tp At time t > 0, our so-calibrated GSR model will produce instantaneous forward volatilities of

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The reader may at this point reasonably ask whether models exist that can produce a time-stationary hump in instantaneous forward volatilities The answer is yes, but such models would generally need more than a single Markov variable to characterize moves in the yield curve We return to this issue in Chapter 12 and, indeed, in many later chapters on multi-factor models

10.1.3 European Option Pricing

In the general one-factor GSR model (10.14), suppose that we fix the mean

reversion function x(t) exogenously, e.g based on empirical observations or from observation of typical decay speed of implied volatilities with option

maturity The function Y(t) in (10.14) is then uniquely fixed by the initial

forward curve, so to complete the specification of the model it remains to determine the function o,(t) In pricing applications, this function is normally found by calibration of the GSR model to observed prices of liquid European options, such as caps and swaptions While we shall postpone most of the intricacies of volatility calibration to later chapters, it should be clear that for a calibration to caps and swaptions to be efficient, we need computationally efficient methods for the valuation of these instruments

In Section 4.5.1, we showed that for any Gaussian HJM model — whether the short rate is Markov or not — caplets can be priced by simple Black- Scholes formulas; see Proposition 4.5.2 for the details Consequently, we here focus our attention on the pricing of swaptions For concreteness, consider a payer swaption expiring at time 79, with the underlying swap paying an annualized coupon ¢ at times 7, < To < < Ty, with 7) > To We recall from Chapter 5 that the swaption payout at time 76 is

N-1 +

Vewaption(Zo) = ( — PŒ\,TwN) —e Ð` nP( Ten) » 7% = đá+t T— Hạ,

+=Ð

(10.21)

10.1.2.1 The Jamshidian Decomposition

Our first approach is exact, and is based on a method developed by Jamshid- ian [1989] The basic idea is to rewrite the swaption payout from an option on a sum of discount bonds to a sum of options on discount bonds To

develop the idea in detail, let us write P(7o,7Tn) = P(To,Tn,x(Zo)) to recognize the dependence of P(Zo, Tn) on x(Zo) = r(To) — f(0, To) through the reconstitution formula (10.18) We also define a “critical” value x* for

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10.1 “The One-Factor Gaussian Short Rate Model 421 which the swap at time Jo is exactly zero; x* can be found by numerical root search on the equation N-1 PŒN,TN,#") +e À ` 74P(TM, Ti41,2*) = 1 (10.22) +=0 Finally, define “strikes” K, = P(To, T;, x"), 4=1, ,N; it follows that N-1 Kyte) Kiqi =1 (10.23) i=0

We are now ready to apply the Jamshidian “trick” Inspection of (10.18)

shows that all zero-coupon bonds P(7,7T;,x(%o)) are monotonically de-

creasing in z(7o), whereby the swaption only pays out a positive amount if

x(To) > x* That is, Vewaption (Zo) N—1 = ( — P(T9,Tn,x(To)) —e »_ rP(To Fen 2(T0) L(2(T))>2*} i=0 N-1 = [i +€ S_ TC; 1ì — P(To, Tn, x{To)) ¿=0 N-1 —C > nP(i,fia,z(0))) L{z(To)>e*}› ¿=0

where the second equality follows from (10.23) Thus,

Všwaption (70) — (TÊN — P(To, Tn, z(1ọ))) ]{xz(Tu)>z*} N-1 +e >> 4 (Kiser — P(To,Ti41,2(To))) lam) >0°} ¿=0 = (KN — P(To,Tw,z(Te)))” N—1 +e) > ri (Kiai — P(To, Titi, 2(To)))* , (10.24) ¿=0

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10.1.3.2 Gaussian Swap Rate Approximation

While the Jamshidian approach above is perfectly adequate for many applications, its use of numerical root search and the need to price a potentially large amount of zero-coupon options can be cumbersome One may wonder, then, whether perhaps a simpler approach is possible, given the simplicity of the dynamics of rates in the GSR framework One obvious idea is to examine the SDE of the forward swap rate in an appropriate annuity measure, introducing approximations as needed to make the dynamics tractable This idea shall be used many times in this book, often in combination with sophisticated techniques for simplification of the swap rate SDEs Here, we have more modest aspirations and will be content with a simpler — yet still functional — approach The reader shall consider this section a warm-up exercise for more accurate approximations to come, in particular in Sections 13.1.4 and 13.1.5 that also cover the GSR case

We start by rewriting the swaption payout as

Vswaption (To) — A(T) (S(Zo) — c)”

where A(t) and S(t) are the swap annuity and forward swap rate, respectively, see (5.13)-(5.14):

A(Ð * AswŒ) = Š) sP(,Ti4), 50) ^ So,w(t) = SON ¿=0

Let Q4 be the measure induced by using A(t) as the numeraire, such that

Vewaption (0) = A(0)E4 (($(Tp) — ©)*}, (10.25)

where E4 denotes expectation in measure Q“ To evaluate (10.25), we need

to determine the dynamics of S(t) in Q4 Lemma 4.2.4 establishes that S(t) is a martingale under Q* From the reconstitution formula (10.18) we also know that S(t) and A(t) must be deterministic functions of x(#):

S(t)=S(t,2(t)), A(t) = Ali, x(t),

so from IJto’s lemma

dS(t) = q(t,2(t))o,(t)dW4(t), g(t, x) = oO Ptt,To,x) — P(t, Tn, 2) dx NT! P(t, Ti41,2)

where W4 is a Q4-Brownian motion and where we use (10.18) to express

the P(t, T;)’s as functions of x Evaluating the partial derivatives yields

P(t, To, x)G(t, To) — P(t, Tn, x)G(t, Ty)

a(t, +) — A(t, x)

_ Slt,2) ye 1

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10.1 The One-Factor Gaussian Short Rate Model 423 where we recall that

7

G(t,T) = | e~ lí ZÁ9)45 gay, t

The function g(t,x2) can be experimentally verified to be close to a constant in x-direction so, as a good approximation, we can write

q(t, x(t)) qữ,Z0)), (10.27)

where %(t) is some deterministic proxy for x(t) With this, the option formula

in the Normal model, see Remark 7.2.9, immediately leads to the following lemma:

Lemma 10.1.10 Let Z(t) be a deterministic function of time, and assume

that (10.27) holds Then

Vewaption(0) = A(0) [(9(0) — c) B(d) + Vuy(d)] ,

where

_ S(0)-e Jv

It remains to choose Z(t) An easy choice is to set £(t) = 0, which will

yield good precision if o,(t) is not too high What also works reasonably well is to simply evaluate q(t,Z) at the forward discount bond curve, i.e

replace P(t, 7;,Z) with P(0,7;)/P(0,t) in (10.26) More accurate choices

for Z(t), as well as refinements to the approximation (10.27), are developed in Sections 13.1.4 and 13.1.5 To d ye | q(t, E(t)? op(t)? de (10.28) 0 10.1.4 Swaption Calibration

In a typical application of the model, the European option pricing formulas from Section 10.1.3 are used to calibrate the model, i.e to find the volatility curve o,(t) so as to match the market prices of one or more calibration targets, most often European swaptions

Trang 19

mature on the same date Ty If this is the case, the strip is called the coterminal swaption strip

With the mean reversion x(t) fixed, we can make the important observation that the value of the swaption expiring at time T; depends on the volatility curve o,(s) for s € [0, 7; | only This can be seen most clearly from the formula for v in Lemma 10.1.10, but 1s also evident from the pricing formula (10.24) and the fact that the discount bond reconstitution formula

(10.18) for P(t, T) does not depend on a,(s) for s 2 ý

The special structure of volatility dependence allows us to perform calibration for one swaption at a time, replacing a potentially multi-dimensional optimization problem with a series of one-dimensional root searches Assume that o,(t) is piecewise flat on the maturity erid, with o; denoting the flat

value on [7;,7;+¡] A possible algorithm based on the formula (10.24) would

then work as follows

1 Assume go, -,0,—1 have been found

2 Set the value a, such that the model price of the (7 + 1)-th swaption, 1.e a swaption that expiries at T,41, is equal its market price, by numerically inverting (10.24) for a, while o9, ,@,-1 are kept constant

3 Repeat Step 2 fori =0, ,N — 2

At first glance, it may appear that the pricing formula from Lemma

10.1.10 will give rise to a linear system on Øộ, 7; 9, allowing us to

execute Step 2 above by simple matrix inversion The reality, however, is slightly more complex as the weight functions g(-,-) also depend on

the volatility curve o,(-) through P(t, T’)’s dependence on y(t) in (10.18) Nevertheless, even with the proper update of y(t) through (10.17) in each

step, the inversion in Step 2 above is sunple fare for any one-dimensional root solver Further details can be gleaned froin Section 13.1.7 that discusses volatility calibration for the closely related quasi-Gaussian models

We should note that the volatility calibration scheme above is not guaran- teed to always work: a condition sometimes called a “volatility squeeze” may cause the inversion in Step 2 to fail if the market value of the T;41-expiry swaption is significantly below that of the swaption expiring at T, In practice, market data is rarely extreme enough for this to happen, and somethnes the problem can be cured by increasing the mean reversion speed s(t) Some care must be exercised here, though, as the usage of unrealistically high mean reversions will impact the inter-temporal correlations of the model (see Chapter 13), which may lead to unrealistic prices for exotics options, as discussed in Chapter 18

10.1.5 Finite Difference Methods

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10.1 The One-Factor Gaussian Short Rate Model 425 turn to Monte Carlo applications in Section 10.1.6 Our discussion of both techniques is rather brief; for further analysis and alternatives we simply refer to Chapters 2 and 3

10.1.5.1 PDE and Spatial Boundary Conditions

Our treatment of finite difference methods for the GSR model — and for short rate models in general, see Section 11.3.1 — essentially involves little outside of straightforward applications of schemes from Chapter 2 Still, let us start by noting that the algorithms we describe here nevertheless deviate quite markedly from the somewhat old-fashioned (and often suboptimal) tree-based schemes that abound in the short rate literature, even in recent work

Consider a claim V with the terminal payout V(T) that depends on the discount curve at time 7 As the discount curve at time T can be

reconstituted solely from knowledge of x(T'), we write V(T) = V(T, z(T)) By standard results (see Section 1.8), we write V(t) = V(t,z(t)), where V(t, xz) satisfies the PDE

OV ØV 1 8*V

Op + (y(t) — z(£)+) On + 5 ort) a5 — (x + f(0,t)) V, (10.29) subject to a known terminal (payout) condition for V(T,, x) This PDE can be solved numerically using finite difference methods, e.g the Crank-Nicolson method in Section 2.2

In setting up the finite difference scheme, we require knowledge of spatial boundary conditions in the z-domain In the absence of contractually agreed- upon boundary conditions (as would be the case for e.g barrier options) one possibility is to set 8?V Ox? ov 5.3 =0, (10.30) ao 4= Ztttn L=Linax

as recommended in Section 2.2.2, where imax and Zmin are the grid boundaries ‘he boundaries are typically determined by probabilistic means, e.g

Cmax — E (x(T)) + ay Var (x(T)), Emin = E (x(T)) — Ø% Var (x(T)),

(10.31) for some confidence multiplier a The moments required in this computation

can be found from equations (10.12)—(10.13); see also (10.40)—(10.41)

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426 10 One-Factor Short Rate Models ]

prevent mis-specification errors at the boundary from affecting the solution

at (t,x) = (0,0) This, in turn, implies that significant computational effort

is spent in areas of the z-domain that are probabilistically insignificant One way to improve on this situation is to rely on the PDE itself to generate

boundary conditions, as described earlier in Section 2.2.2 (see also Section 9.4.4) We present the details of this idea in the next section

10.1.5.2 Determining Spatial Boundary Conditions from PDE

We assume that the PDE (10.29) has been discretized on a spatial grid {2}, so that V;(t) = V(t,z,), etc Let us focus on establishing the

boundary condition at 29 = min, say Using a 6-method discretization scheme, as in Section 2.2, with an upward discretization of the x-derivatives

we get, at some time step |f,£ + 6], Vi (t) — Volt) Vo(t + ) -_W@) „ b(t, 20) cL (10.32) + (1-0) lt + 6,2) 20+ 9) = hổ + 9) #1 —- Xo g 2 Vo(t) — Mì) ” Vi (t) — Vo(t) 1 T5 r(t) { #2 — Z mm loop + ——o,(t + 6)” (10.33) | eh 1 #3 — Ly #1 — Z0 5 (x2 — 20)

= 0o + ƒ(0,)) Vo) + (1 — 0) (o + ƒ(0,£ + ô)) Vo( + ô), — (10.34)

where /(£,+) Ê (£) — z(£)z This equation can be rearranged to write Vọ(£) as

Vo(t) = ki(t)Vi(t) + ko(t)Va(t) + go (t+ 4), (10.35)

where k,(t) and k(t) are easily computed functions of the process parameters,

and where go(t + 6) is a function of Vo(t + 6), Vi(t + 6), and Vo(t +5) We leave it to the reader to write out k,, ke, and g in detail Applying similar principles, we get

Vina (t) = Km—1(t)Vm—1(t) + km(t)Vm(t) + gm+i(t+4) (10.36)

Comparing (10.35)-(10.36) with the equations (2.12)-(2.13), we see that the boundary conditions (10.35)-(10.36) can be incorporated into our usual tri-diagonal roll-back scheme by simply interpreting f(t, 70) = go(t + 6) and f(t, 2m41) = 9m4i(t + 6) in the scheme of Section 2.2 As we are rolling

back in time (from t+ 6 to t) when using the finite difference equations,

both go(t + 6) and gm4i(t +46) are known at time t, so this interpretation

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10.1.5.3 Upwinding

For the PDE (10.29), notice that the condition (2.34) states that convection

domination can cause spurious oscillations to creep into the finite difference scheme unless

ly(t) — x(t)alAe < op(t)?, (10.37)

for all x spanned by the finite difference grid Since o,(t)* is typically a small number (around 0.001), it is not uncommon for this inequality to be violated a% the edges of the finite difference grid (i.e in the neighborhoods around ro and Lm+1) where the mean reversion pushes or pulls strongly at z(t) To avoid numerical difficulties with the finite difference scheme, it is therefore recommended to apply the upwinding scheme in Section 2.6.1 While in principle this may reduce the spatial convergence order of the scheme, in practice the effect of upwinding on convergence is often minimal provided

that the finite difference grid is dimensioned in such a way that (10.37) is

only violated in a fairly small portion of the grid

10.1.6 Monte Carlo Simulation 10.1.6.1 Exact Discretization

Consider the problem of pricing a derivative security that pays an amount

V(T) at time T, where V(T’) may be a function of the entire path of the

discount curve over time interval [0,7] Working in the risk-neutral measure, we are thus interested in computing

V(0)=E (VŒ)e" for rode)

= P(0,T)E (V(T; {2(t) :0<t<T})e fe z0)dm) „ (10.38)

where the second equality shifts variables to a(t) = r(t) — f(0,t) and emphasizes the dependence of V(7’) on the entire path of z(t) Recall from

the discussion in connection with Proposition 10.1.7 that there are distinct

advantages to working with the variable x(t) = r(t) — f(0,t) rather than r(t) In the GSR, model, the dynamics for x(t) are given by (10.16), i.e

dx(t) = (y(t) — x(t)z(t)) dt +o,(t)dW(t), y(t) = | en 2S, dus Cy)? du,

For the purpose of Monte Carlo pricing of (10.38), we discretize the time-interval into a schedule ty < t) < < ty, with tp) =O andty = T The exact choice of the schedule depends on the particulars of the payout V(T); if, say, V(T) only depends on the yield curve at time 7, it suffices to

set N = 1 Now, we can solve the Gaussian SDE for x(t) (see Section 1.6)

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428 10 One-Factor Short Rate Models ]

toad titi trod xa) = eC để” #09482 (0) + / ch” xu) đg ti ti4d tự tÍ ew fe (u)du 5 (5) dW(s), (10.39) t; which we recognize as being a Gaussian random variable with moments t; (¿+1 tj E (x(te41)|a(ts)) =e" ode (2) + / eT Đề) 4M (8) đc, ti (10.40) bid ta 2 Var(e(ii)le(M)) = |” (6C #92692, (6))ˆ a; t¡ (10.41) Advancement of z(£) on the schedule can thus be đone in bias-free manner, by writing

#i+i) = B(i+i)|lz(/)) + VVar (e(0+i)|())2i, 2=0, ,N T1, wher€ Zo, , w—1 Ìs a sequence of independent standard Gaussian random variables

For every date on the simulated path x(to), ., x(tn), we can use

the reconstitution formula in Proposition 10.1.7 to reconstitute the entire discount curve, in turn allowing us to compute V(T) on the path To evaluate

(10.38), it remains to simulate the quantity I(T) = — fee

on the path Given x(t9), ., x(ty), an obvious choice would be to com-

pute /(T’) by quadrature (e.g trapezoidal integration, or similar) As this inevitably introduces a discretization bias (see Andersen and Boyle [2000]

for more analysis), it is preferable to use the following result

Trang 24

Also, we have

Cov (x(tigi), (tii) Li );36h))

Proof Straightforward but tedious calculations for Gaussian random variables ©

Over the time step {t;,t;41] we advance /(t) according to the formula

I (tigi) = E (T(tigs)|T (ta), 2(ta)) + / Var T(t i), (ta) Zi,

where Zo, Leng Z N—1 is a sequence of independent standard Gaussian random variables, and where the required moments of I(t;41) can be found in Lemma

10.1.11 To honor the covariance between the x(t) and I(t) processes, we

require that the variables Z; and Z, are correlated:

Ti = #(z+1)|T(;), z@¿)) |

J Var (I (tia) T(E: t;))«/ Var (x(tia1) |Z (ti), x(te)) As explained in Chapter 3, correlated Gaussian samples can be generated from uncorrelated samples through the Cholesky decomposition

The scheme outlined above allows us to simulate bias-free paths of the

variables x(t) and I(t), which in turn allows us to compute independent, unbiased samples of V(T)e/(7) Monte Carlo estimation of the expectation

for V(0) can then be performed in standard Monte Carlo fashion, by forming sample averages The discretization scheme involves several time-integrals over dates in the observation schedule, many of them nested; it goes without saying that these integrals should be pre-computed before actual path simulations commence Corr (2:2 Zi; i) = 10.1.6.2 Approximate Discretization

For a quick-and-dirty implementation of the Gaussian model, we may elect to skip the algorithm in the previous section and instead apply one of the approximate discretization schemes in Section 3.2 As a starting point, we have the vector SDE

(G8) = (=a) (8a

A plain-vanilla Euler scheme from Section 3.2.3 would write

(Fa | _ Kì + (ue “A ) A, + (oy) Z,VA,,

Trang 25

where A; = t;,; ~¢t,; and Z, is a standard Gaussian random variable Unless x is small, this scheme cannot be recommended due to the stability issues discussed in Section 3.2.3 As explained in Section 3.2.3.1, it is preferable to incorporate the fact that

; : _ ~ ft" selu)du,, nhi — zz(u)du E.(z(,+1)|zŒ,)) =e 2" z(,) + € y(s) ds t, 1— ez¿)Œi+i—t) x(t)

we @ AG Metab ace.) 4 y(t;)

That is, we write

w(t, jA;

#(¿+1) ^ — e~Z)3:~(#,) + ay _ ti) \ oN) Zi

Cie) = ( (+1) 1(t¡) - ®(,)A; v4

This scheme has first-order (weak) convergence Higher-order schemes can be found in Section 3.2, but are essentially obsolete here: if truly low bias is critically important, we should use the unbiased scheme from Section 10.1.6.1

10.1.6.3 Using other Measures for Simulation

The need to simulate J(¢) can be avoided entirely by a suitable change of the probability measure Switching to the terminal measure Q? (see Section

4.2.4), we rewrite (10.38) as

V(0) = P(0,T)E* (V(T)),

aud observe that we now need to simulate a(t) only in order to calculate the payoff The dynainics of x(t) under the terminal measure Q? follow from

(4.34),

da(t) = (y(t) ~ x(t)a(t)) dt + op(t) (dW (t) ~ op(t, T) dt) (u(t) — o,(t)°G(t, 7) — x(t)a(t)) dt + op(t) dwT(t),

|

with W*(t) being a Q?-Brownian motion The dynamics remain Gaussian and Markov under the terminal measure, and hence x(t) can be simulated bias-free on the time grid {t;}%_,

An alternative to the terminal measure that is “closer” in some ways to the risk-neutral measure, yet still allows one to avoid the simulation of

I(t), is the spot measure from Section 4.2.3 We recall that this measure is

associated with the discretely compounded money market account B(t),

i

B(t) = P(t, ti+1) H P : f€( tàn]

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10.2 The Affine One-Factor Model 431

Under the spot measure Q”, we obtain

N-1

V(0) =E* CT Pnstara(a) WC]

n=0

where we explicitly indicated the dependence of the discount bond on the state process x(t) Notice that the random variable under the expectation

operator is a function of x(t) on the grid {t;}*_, only Moreover, over the

interval |t,,tn+1], the measure Q* coincides with the Tn+1-florward measure,

which gives us the dynamics of x(¢),

dx(t) — (y(t) _ ơr(t)?G (t, tn41) x(t)a(t)) dt

+or(tdW(t), t€ (tnstn4al,

with W(t) a Q?-Brownian motion Again, we can generate a sample of

t(tn+1) from x(t,,) in a bias-free manner We refer the reader to Chapter 14

for more details on numeraire simulation strategies

10.2 The Affine One-Factor Model

Earlier (in Section 10.1.1.3) we identified the non-zero probability of negative

interest rates as one of the drawbacks of the one-factor GSR model Another problem is the lack of interest rate dependence in the GSR short rate volatility, leaving the user with no means of controlling the volatility skew implied by the model While there are different ways of addressing these issues, oue type of model that can, in part at least, address both of these shortcomings of the GSR model is the affine short rate model This model — or, rather, model class — constitutes a significant extension of the GSR model (which in fact is a member of the affine class), yet retains a high degree of analytical tractability Originally introduced by Duffie and Kan [1996], the affine class of short rate models enjoys high popularity among practitioners and academics alike, particularly for econometric work The affine models are also quite useful for derivatives pricing, although ultimately the constraints one need to impose on diffusion dynamics can be too strong for some applications

10.2.1 Basic Definitions 10.2.1.1 SDE

Consider a time-homogeneous one-factor short rate process of the form

Trang 27

where W(t) is a Brownian motion in the risk-neutral measure, # > 0, a > 0,

and ở are constants, and v(-) : R > R is a deterministic function of the level of the short rate We notice that the drift of (10.44) is affine, i.e linear, in r(t) If the square of the diffusion term in (10.44) is also affine, we say that

(10.44) is a time-homogeneous affine one-factor short rate model Evidently,

the function u(r) is thus limited to the form

u(r) = Vac Br, (10.45)

for constants a and 6 We notice that the special case of 6 = 0 produces the GSR model of Section 10.1.2.2, whereas the case a = 0 produces a square-root type model similar to those encountered, for stochastic volatility modeling, in Chapter 8 The case a = 0, 8 = 1 was first studied by Cox

et al [1985] and is known as the Coz-Ingersol-Ross (CIR) model

10.2.1.2 Regularity Issues

Not all combinations of parameters in (10.44) and (10.45) produce a well-

defined SDE If 6 = 0 for all t, we must require that a > 0 for all ¢ to ensure

that u(r(t)) is defined If 6 #0, for the square root in (10.45) to exist, we must ensure that the drift term in (10.44) has the same sign as 6 whenever a+ 6r(t) = 0 That is,

xB(0+a/B)>0, 8#0, (10.46)

for all t > 0 Notice that if we wish for the volatility term in (10.45) to be strictly positive (a + Sr(t) > 0), we need to replace this condition with the stronger Feller condition (recall Proposition 8.3.1)

z8 ( +œ/8) > 2

For the CIR model the requirement that r(t) stays strictly positive can be seen to translate into the classical condition 229 > o?

For the purposes of modeling interest rates, it is most reasonable to assume that x > 0 to ensure that rates are mean-reverting rather than mean-fleeing, and that 6 > 0 In this case, the domain of the short rate becomes

r(t) € [-a/B,oo), 8>0, (10.47)

and r(t) € (—0oo, 00) for the case 8 = 0, a > 0 (Gaussian model) Evidently,

to keep r(t) non-negative for all t, we need to set a < 0, subject to the

restriction that —a/B < r(0)

10.2.1.8 Volatility Skew

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10.2 ‘Vhe Affine One-Factor Model 433

model can generate skews ranging from a Gaussian process (a > Ø) to a square-root process (a < £) In the usual language, for non-negative a, 8, the skew “power” of the affine model thus ranges from 0 to 0.5 By allowing a to be negative, effective skew powers above 0.5 are possible, although the allowed range of the underlying process r(t) will then be floored at some positive level, which may have undesirable side effects if a/( is not close to zero

10.2.1.4 Time-Dependent Parameters

The SDE (10.44) does not depend on time and hence will generally not match the initial yield curve at time 0 As we did for the Gaussian model,

we may extend the SDE to have time-dependent parameters, e.g

dr(t) = s(t) (0(t) — r(t)) dt + o(t)\/a + 8r() dW(t) (10.48)

Notice that we have not introduced time-dependence in @ and /, leaving the domain (10.47) unchanged® Not all functions x, 0,0 produce a well-defined

SDE; for instance, if x(t) is positive (which is always the case in practice) and 6 > 0, then (10.46) shows that we need

x(t) Bd(t) > —a (10.49)

in order for (10.48) to be well-defined

Remark 10.2.1 For generality we allow x(t) to be a function of time through-

out As argued in Section 10.1.2.3, however, it is often most reasonable to

let s(t) be a constant

10.2.2 Discount Bond Pricing and Extended Transform

Starting from the time-dependent SDE (10.48), we now turn to the search

for a discount bond reconstitution formula, i.e a formula that allows us to compute the risk-neutral expectation

P(t,T) = E; (e- dc na) | (10.50)

as an explicit function of r(t) Rather than directly attacking (10.50), we

turn to the more general problem of establishing the so-called extended transform g(t, T;¢1,c2) defined by

g(t, T3¢1, co) = Ey tu J rau) , C1, €C (10.51)

Trang 29

Notice how this generalizes the idea of the moment-generating function from Chapters 8 and 9 Also note that the knowledge of g allows us to find discount bond prices as a special case,

P(t, T) = g(t, T;0, 1)

For the values of c; and cz for which g exists, we can use the following result, which is an extension of Proposition 9.1.2

Proposition 10.2.2 For the model (10.48), whenever the extended trans-

form g in (10.51) is defined, it is given by

g(t, T3c1,¢2) = exp (A(t, T; c1, co) — Bt, T3 cr, co)r(t)), (10.52)

where A and B satisfy a system of Riccats ODEs 1

= — x(t)v(t)B + 22) °aB” = 0, (10.53) = + x(t)B+ 5o(t)° BB = C2, (10.54)

subject to the terminal conditions A(T;T,c1,¢c2) = 0, B(T;T,c1,c2) = c4 Proof Follows that of Proposition 9.1.2 closely O

Remark 10.2.3 If the parameters a and £ are functions of time, Proposition

10.2.2 continues to hold if we simply replace a, @ with a(t), 6(t) in the

Riccati equations for A and B

Proposition 10.2.2 establishes that the joint characteristic function of

r(T) and f r(u) du is known analytically for the affine model, a result that

accounts for much of its popularity in the financial literature Solution of the

Riccati equations (10.53)—(10.54) can be done quickly and robustly by any number of standard ODE schemes, such as the Runge-Kutta method (see

Press et al [1992]) For the case where parameters are piecewise constant in time, establishing A and B in Proposition 10.2.2 can also be done analytically; see Section 10.2.2.1 below

10.2.2.1 Constant Parameters

We now turn to establishing the extended transform g for the special case where all parameters in (10.48) are constants As a warm-up case, we first list a result for the CIR case

Proposition 10.2.4 Consider the CIR model

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10.2 The Affine One-Factor Model 435

œnd let g(t, T1; e\, ca) be defined as in (10.51) Set 7 = 7 (62,0) = Vz? + 2ø Then g(t, T; c1,¢2) = exp (Acir(t, 73 0,0, ¢1,¢2) — Barr (t, 7; 0,0, ¢1, c2)r(€)), where Acir(t, 7; 9,9, 61,2) = #007? (# + (e2,0)) (T —t) (sc + +(ca,đ) — cơ?) (czz;z)—9 — 2) — 2z0ø7?In |1+ ( 27 (c2,0) and Borr(t, 7; 0,0,¢1,¢2) =

(2cs — zei) (1L — e266292)—1)) + + (co, 0) er (1+ eM)

(+ + Y (ca, Ø) + cơ?) (1 — e~*(es,ø)—1)) + 2*% (ca, ơ) e~7(c2,ø)(T'~t) ,

Proof The result is a small extension of Proposition 8.3.8, and follows by

direct solution of the ODEs (10.53)—(10.54) O

Armed with this result, it is straightforward to extend it to the general constant-parameter case

dr(t)=x(0—r(t))dt+oV/at+Pr(t)dW(t), B>0 (10.56) In particular, we notice that if y(t) = a+ @r(t), then y(t) follows the SDE

dy(t) = Bdr(t) = (G0 +a—y(t)) dt + Borv/y(t) dW(Z),

which is of the form (10.55) We also have

g(t, T; c1, Co) = Ey [se (anit) — Cạ ƒ sa)

“8 bề Le) 3 [ (%°)4)) g te TẾ |

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Lemma 10.2.5 The extended transform for the constant parameter affine model (10.56) is g(t,T;¢1,c2) = exp (A(t, T; c1,c2) — BUt, T; e1, c2)r(t)), where A(,Tìei, œ) = cia/8 + ca ~ t)/B + Acir (: T; BO + 0, Bo, 7 2) —aBorr (+ T; 80 +, Bo, 3 3):

B(t,T;¢c1,c2) = BBcir ụ TBO + a, Po, 2 2)

and the functions Acin and Bor are given on Proposition 10.2.4

10.2.2.2 Piecewise Constant Parameters

We can use the results established in Section 10.2.2.1 to compute extended transforms for the case where we are given a time grid 0 = tp <f) <ta< on which all model parameters x and a can be assumed piecewise constant The resulting recursive routine is a robust and efficient? alternative to

Runge-Kutta solvers

For simplicity of notation, let us define g(ti,t;;¢1,c2) = gi,j (C1, C2), A(ti,tj3€1,€2) = Azj (er, 2); and so on Then, from Proposition 10.2.2,

Gig (Cis C2) = Arica “PEI Bi ere) 5 > 4, (10.57)

and, using the law of iterated conditional expectations,

gi-1,7(€1,€2) = Be, (oor mer fi oa)

<= Et, _, [E ew fil, re) )

— Ke, [eet Je tu up Cua fi) roa) )

f

—cs [,' „ r{u)du

— Đụ (« 1 øi¿(ei:e3)]

Inserting (10.57) into the last equation then yields

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10.2 The Affine One-Factor Model A37 fj Gr—1,9 (C1 C2) — Hạ, (5 HH "eld ps fener MB: (ne) — e0 (9199) 0 1 (By; (C1, C2) , C2) Applying (10.57) to the right-hand side of this equation leads to Aj] j(e1,¢2)—-r(t,-1)B,—1.,(¢1,¢2) e — Aes (61,62) pAi—1 ( Bij (1,62) ,c2) —r(ti— 1) Bi-1.i( Bu j (61 s¢2),€2) or, finally,

Aj-1,j (C1, €2) =Ai,y (C1,€2) + Ai—14 (Big (€1, C2) , 2), (10.58) Bi-1,9(€1, C2) =Bi-1,; (Bij (C1, ¢2) , €2) (10.59)

As paraineters are constant on the time grid, the functions A;_1,; and By_1; can be computed in closed form from the results of Lemma 10.2.5 For a

fixed 7, (10.58)-(10.59) can be used in backward fashion to establish A; ;

and 6; ; Íor ¿ = 7 — 1,7 — 2, ,0; the recursion starts with an application of Lemma 10.2.5 to compute Aj;_1,; and B;_1,;

10.2.3 Discount Bond Calibration 10.2.3.1 Change of Variables

In the affine SDE (10.48), the role of the mean reversion level v(t) is to

calibrate the model to the initial term structure of discount bonds As

we discussed in the context of the GSR model, Y(t) will depend on the

derivative 0f(0,t)/0t which may, for many curve construction algorithms, be irregular For practical applications of affine models, it is therefore strongly recommended to follow the advice of Section 10.1.2.2 and rewrite the model

in terms of a variable that measures the difference between r(t) and f(0,t) Let this variable be x(t), defined as

x(t) = r(t) — f(0,t) The SDE for z(t) becomes

da(t) =dr(t) — CHO i

~ (w(t) — x(t)x(t)) dt + o(t) (EQ) + Beh dW(t), (10.60)

where x(0) = 0, €(t) =a+ 8f(0,t), and

a(t) = x(t)0(t) — Of (0, t)/Ot — x(t) f(0, t)

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Written in terms of x(t), the extended transform in Proposition 10.2.2 becomes g(t, Ts C1, C2) =e OP) fi f(O,u)dup leTszữ)=e fe (udu) P(0,T)° =eT°1/(0,7) ae exp(C(E, L; Cl; Ca) a x(t) B(t, +; C1, c2)), (10.61)

where B solves (10.54) and C can, after suitable translation of the results

in Proposition 10.2.2 to the process (10.60), be written as the solution to

the Riccati ODE:

dC 1 2

=, 7 w(t)B + s7) 4(B” =0 (10.62)

10.2.3.2 Algorithm for w(t)

We now assume (but see Section 10.2.5) that a and § have been fixed, and that s(t) and a(t) are known for all values of t > 0 In the SDE (10.60) for z(t), it only remains to establish the function w(t), which shall be done to

match observed discount bond prices at time 0

To make matters more concise, let us set b(t,T) = B(t,T;0,1) and

c(t, ) = C(t,T;0,1) such that, from the definition of C(t, T), P(0,T P(t,T) = 9 (t,T;0,1) = Day Men (10.63) The functions 6 and c obviously satisfy a Riccati system, de I 2 2 _ db 1 2 ap2 _ — + z()b + 27) 6b“ = 1, (10.65) where c(T,T) = 0(T,T) = 0

Setting ¢ = 0 in equation (10.63) establishes the fundamental calibration requirement that c(0,7) = c(0,7;w(-)) = 0 for all T which, combined with

(10.64), defines a so-called Volterra integral equation for w(-) We can solve it on a time grid to < ty < te < < ty by iterative bootstrapping of the

equation c(0,t;;w(-)) = 0 Assuming that w(-) is piecewise constant at a level w; over the time bucket (t;,t;41], we can use the following algorithm

1 As a pre-processing step, find b(t;,¢;) for all i,j, j > 7, by solving (10.65)

This does not depend on w(-)

2 For a given 2, assume that w, is known for 7 < 0

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10.2 Phe Afine One-Factor Model 439 4 Compute w, as the solution to O(t,;) — ứ¿ feo b(s,tj41)ds =0 5 Repeat steps 2—4 for all i =0,1, ,N—1

Notice that no numerical root search is needed and that the computational

complexity of the scheme is O(N) By modifying Steps 3 and 4, other

interpolation techniques can be supported, although stability issues might

come into play See also Press et al [1992] for more general schemes to solve

Volterra equations

We should note that there may be cases where the algorithm above will fail, in the sense that the basic regularity condition (10.49) will prevent a valid solution for w(-) from existing This is a fundamental issue with non-Gaussian affine short rate models, but is rarely observed as very strongly downward-sloping yield curves are required to trigger the problem (see the

discussion in Hull and White [1994a))

10.2.4 European Option Pricing

The short rate volatility function o(t) in the affine model (10.60) will normally

be determined through calibration against swaptions and caps/floors For such calibration to be computationally feasible it is, of course, important to establish fast methods for pricing European interest rate options

kor simple options such as caplets or, equivalently, options on zero- coupon bonds, the availability of the moment-generating function for the logarithm of the bond (see Proposition 10.2.2) allows for application of the Fourier methods?® of Section 8.4 Extensions to swaption pricing through the Jamshidian approach of Section 10.1.3.1 is possible in principle, but the need to perform Fourier integration of a large number of Riccati ODE solutions makes this approach impractical Several approximation techniques have been proposed in the literature; see, for instance, Collin-Dufresne and Goldstein [2002a] for a survey and details on a method based on Gram- Charlier expansions Our preferred approach to swaption pricing in the affne model borrows the techniques of Section 10.1.3.2 to work out an approximation for the swap rate martingale dynamics We shall outline one straightforward and quite accurate approach here; as was the case for the GSR model, we again will stop short of the full-blown projection techniques that will be introduced later in this book for more realistic candidates for actual trading applications

Let us, as in Section 10.1.3.2, start out by rewriting the swaption payout

as

Vewaption(To) = A(To) (S(To) — 0) , (10.66)

where

*°For time-homogeneous models, closed-form pricing formulas for options on discount bonds exist for some models, including the CIR model (see Cox et al

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N-1

~ » T¡P, 1+1); S(t) = aE

i=0

Let Q4 be the measure induced by using A(t) as the numeraire; in this

measure S(t) is a martingale By the reconstitution result (10.63) we have

dS(t) = oe) (t)/E(t) + Ba(t) dWA(t) (10.67)

where WA(t) is a ¬ motion and

OS(t) _ b(t, To) P(t, To) = b(t, Tv) P(t, Tn) i) A(t) _= T;b(t, 1++1)PŒ, Ti+1): ¿=0 sẽ

The dynamics (10.67) are generally intractable, but S(t) can — as was

the case for the GSR model — be verified to often be well approximated by a linear function of x(t), with slope and intercept being functions of time

Using a Taylor expansion around some point Z (e.g ZT = 0, but see the discussion in Section 10.1.3.2), we can find C(t), x(t) such that

S(t) = C(t) + x()x(t),

and then (10.67) approximately reduces to an affine SDE for S(t):

dS(t) & x(t)o(t) VE(t) + Ba(t) dW (t)

_ (t) — C(t) A

= x()z( neo +8 (SSO) awa

= a(t)./é.(t) + Bo(t) S(t) dW“ (t) (10.68)

While valuation of the payout (10.66) cannot be accomplished in closed

form when S(t) follows the time-dependent affine SDE (10.68), we can

always rely on transform-based methods Indeed, it is evident that the characteristic function of S(Zo) can be constructed by applying Proposition

10.2.2 and Remark 10.2.3 to (10.68), whereafter Theorem 8.4.3 gives us a

way to calculate the required expected value in

V(0) = A(0)E9° ((sŒ) _ e)*) | (10.69)

We trust that the reader can see how this would work, so we omit the details Instead, we proceed to further simplify matters, through time averaging of parameters

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10.2 The Affine One-Factor Model 44]

dS(t) = ø()W 8:() + S(t) dWA(t), (10.70)

where w~ is a some constant One approach for setting ~ is to simply match

quadratic variance of S(t) over |0, To], i.e To Tạ / a(t)? B,(t)a dt = / z()28,()£;(® dt 0 0 _ J6" ø()°8,(0)6;(0) di: f" ø0)28;(0) at

A more sophisticated alternative would be to rely on a small-noise expansion,

as in Chapter 7 In any case, for the SDE (10.70), the expectation in (10.69)

can be evaluated in closed form To see this, simply define y(t) = ~ + S(t) and note that

dy(t) = a(t) /Bs(t)V y(t) dWA(t), y(0) = b + S(0), (10.71)

p

and

V(0) = A()E ((S(Zo) —c)*) = A(OE®" ((y(To) — cy)*), (1972)

with cy = w +c Since y(t) in (10.71) is simply a (time-dependent) CEV

process with CEV power 1/2, computation of the call option expectation in

(10.72) can be carried out by the formulas in Section 7.2 Swaption prices

produced this way are, in our experience, accurate and robust, and much more convenient to compute than by competing methods

10.2.5 Swaption Calibration

As we showed in Section 10.1.4, calibration of the GSR model volatility to swaption prices is a matter of straightforward bootstrapping Unfortunately, matters are more complicated for general affine models

10.2.5.1 Basic Problem

To gain insight, let us first consider the simple problem of calibrating the

model volatility function a(t) in (10.60) to match the time 0 price of a

A-tenor zero-coupon bond option maturing at T Assuming that the initial

yield curve is known at time 0, how much volatility information is needed to

price this option? The answer to this question depends on the specification of £ and ổ

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(10.63) depends on the initial discount curve all the way to time T+ A, it

only requires the specification of a(t) to time T Further, the state of «(T)

only depends on a(t), § << LT In total, when 6 = 0, the discount bond option

payout is only affected by {a(t)}o<i<r, irrespective of the magnitude of the

bond tenor A This is also obvious from the reconstitution formula (10.18)

If 6 40, however, we see from (10.65) that b(T, T + A) depends** on the volatility {o(t) }ocecr+a- This again makes c(t, 7 +A) depend on volatilities in [t,7 + Al, requiring the full knowledge of {ø(f)}o<i<r+aA to price the

option at time 0 This fact has implications for calibration to, say, swaption prices as regular bootstrapping techniques cannot be employed

10.2.5.2 Calibration Algorithm

Consider now the situation where we wish to calibrate our volatility function g(t) to a swaption strip defined on a maturity grid 0 = 7p < Ti < < IN Recall that a swaption strip consists of N — 1 swaptions expiring at times T;,i=1, ,N— 1; we here assume that all swaptions are written on swaps that mature at time Ty (coterminal strip) According to the discussion above, pricing any one of these swaptions — even the short-dated ones —

aan affine model will require knowledge of {o(t)}o<tery- As it would be

too slow to calibrate volatilities by simultaneous, imulti-dimensional root search on all levels o{T;), 7 = 0,1, ,.N, we instead notice that while, say, the swaption maturing on date T, depends on volatilities everywhere on

(0, Ty}, its dependence on the volatilities in I0, 7] is much stronger than on the volatilities in the interval (T;, 7] Assuming that a(t) is piecewise

constant on the maturity grid — with o; denoting the flat value on (T;, Tis] — we can use this observation to propose the following iterative calibration

approach

1 Start out by setting all ơ¿,?=0, ,A =1, equal to a reasonably chosen constant, or equal to values approximated from a calibrated GSR!” model

9 Compute w(-) to match time 0 prices of the N discount bonds maturing on 1,75, , Ÿy One can use the algorithm in Section 10.2.3.2 for this 3 Set the value g9 — but leave all other volatilities oj, 7 = 1, ,N — Ì,

unchanged — such that the swaption maturing at time 7; is priced correctly We can use the pricing techniques in Section 10.2.4 for this 4 Repeat Step 3 for 71, 09, .,7N—2, always leaving future (but not past)

points on the volatility curve unchanged

Repeat Step 2 and recompute all swaption prices

6 Repeat Steps 3-5 until all swaptions are priced within given tolerances

crt

11 Recall that we solve b(t,7' + A) backward in time from the known boundary condition at t= f + a

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10.2 The Affine One-Factor Model 443 Notice that in Step 4, altering o; will slightly distort the prices of swaptions maturing at dates earlier than 7j;; this necessitates the iteration in Step 6

We (re-)emphasize that the algorithm above, when applied to the Gaus-

sian model, will converge within one iteration in Step 6 Finally, we note that the calibrated model needs to be checked against the regularity conditions discussed in Section 10.2.1.2; if conditions are violated, the problem may potentially be remedied by increasing a

10.2.6 Quadratic One-Factor Model

In conclusion, let us consider an interesting special case of an affine class A quadratic Gaussian one-factor model is obtained by specifying the short rate to be a quadratic function of a linear Gaussian process,

r(t) = a(t) + B(t)y@) + y(t)yt)’, (10.73) where

dy(t) = —x(t)y(t)dt + o(t)dW(t), y(0) =0 (10.74)

While this is not immediately obvious, the model is indeed of affine type,

albeit in two factors If we denote u(t) = y(t)*, we see that r(t) is a linear function of the state vector (y(t), u(t)), which follows the SDEs

4 (0) = (, | ¬— ») dt +o(t) ( ` ») dwt), (1075)

which is affine

We consider multi-dimensional quadratic Gaussian models in a fair amount of detail in Chapter 12, so we shall be suitably brief here The affine connection makes it unsurprising that bond reconstruction formulas exist for the quadratic model In fact, we have that zero-coupon discount bonds

are exponentials of a quadratic function of y(t),

P(t,T) = P(t,T;w()) = c+~907)90)=e7)90)

with the coefficients a, b, c satisfying Riccati ODEs

In soine ways the parameterization (10.73)—(10.74) is more convenient than the general affine specification For example, with the discount bonds

known functions of a Gaussian factor y(t), a swap rate — or a swap value —

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10.2.7 Numerical Methods for the Affine Short Rate Model Much of the material on numerical methods for the GSR model applies to the affine short rate processes, so we shall be brief Turning first to finite difference methods, let us again emphasize that the spatial variable should

be set to be x(t) = r(t) — f(0,t) rather than r(¢) itself The dynamics for x(t) can be found in (10.60) and lead to the general derivatives pricing PDE

2

To + (ult) — 2elt)e) + Sole)? (E(t) + 8ø) 5 = (ø + ƒ(0,9)9/

which can be solved by standard methods, given appropriate terminal and boundary conditions We refer to Section 10.1.5 for general guidelines Dimensioning of the spatial dimensions of the finite difference grid by probabilistic means will require estimates for the mean and variance of x{(T), with T being the terminal horizon We can compute these from the moment-generating functions established earlier, or, perhaps more easily, by approximating the SDE for x(t) as being approximately Gaussian If r(¢) is close to a CIR process, we can also use the analytical moment results established in Corollary 8.3.3 When establishing the terminal boundary

function (i.e the option payout), we can rely on the reconstitution formulas in (10.63) to turn values of x in the finite difference lattice into the discount

bond prices that are required to evaluate the payout

As for Monte Carlo methods, many of the principles of Section 10.1.6 continue to apply, and we can draw on material in Chapter 8 to design schemes to advance x(t) through time To elaborate a bit on this, suppose that we are interested in advancing x(t) from time ¢; to time t;4, Assume that all parameters in (10.60) are piecewise constant, such that

where! 2 — z(t¡), Œ¿ — (j0(t¿) /2z(t;), ỚØ¿ — o(t,), and E; = E(t; ) Defining

Ei)/8, where

dy(t) = 2; (Bqi + & — y(t)) dt + Bor/y(t)dW(t), y(ts) = & + 6x0)

(10.76) Simulation of this SDE, however, was discussed in detail in Section 9.5 where a number of practical algorithms were introduced We should notice that typical parameterizations of (10.76) will rarely violate the Feller condition, making this SDE considerably easier to deal with numerically than the stochastic volatility applications in Section 9.5 Additional material on Monte Carlo simulation of generic short rate processes — most of which also applies to affine processes — can be found in Section 11.3.3

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11

One-Factor Short Rate Models II

While the affine specification (including the Gaussian case) of Chapter 10 is, without doubt, the most popular one-factor short rate model in practice, quite a few other models have been proposed in the literature In this chapter we cover the most important of these models, paying special attention to the case where the short rate is log-normal We also briefly discuss some issues in the econometric estimation of short rate models, and introduce the important concept of unspanned stochastic volatility

As most of the models introduced in this chapter have no analytical bond reconstitution formulas, their calibration to the initial term structure requires numerical work Accordingly, the second half of this chapter is dedicated to numerical methods for pricing and, especially, calibration of models based on generic short rate SDE Particularly useful in this regard is the discussion in Section 11.3.2 on efficient finite difference schemes based on the important concept of forward induction

11.1 Log-Normal Short Rate Models

Given the pervasiveness of the log-normal Black-Scholes model in derivatives pricing theory, it should come as no surprise that many authors have attempted to specify one-factor short rate models where the dynamics of

r(t) are of the form dr(t) = O(dt)+o,(t)r(t) dW(t) with deterministic o,(t)

This section reviews this class of models which, somewhat surprisingly, turns out to have a number of rather severe drawbacks

11.1.1 The Black-Derman-Toy Model

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